A combinatorial theorem in group theory

Abstract:

There is an anti-Ramsey theorem for inhomogeneous linear equations over a field, which is essentially due to R. Rado [2]. This theorem is generalized to groups to get sharper quantitative and qualitative results. For example, it is shown that for any Abelian group A (written additively) and any mappings ${f_1}, \cdots ,{f_n}$ of A into itself there exists a k-coloring $\chi$ of A so that the inhomogeneous equation \[ \sum \limits _{i = 1}^n {({f_i}({x_i}) - {f_i}({y_i})) = b,\quad b \ne 0} \] has no solutions ${x_i},{y_i}$ with $\chi ({x_i}) = \chi ({y_i})$ for all $i = 1, \cdots ,n$. Here the number of colors k can be chosen bounded by ${(3n)^{n - 1}}$ which depends on n alone and not on the ${f_i}$ or b. For non-Abelian groups an analogous qualitative result is proven when b is "residually compact". Applications to anti-Ramsey results in Euclidean geometry are given.